DISSERTATION PAPER Modeling and Forecasting the Volatility of the EUR/ROL Exchange Rate Using GARCH Models.

1
DISSERTATION PAPERModeling and Forecasting the
Volatility of the EUR/ROL Exchange Rate Using
GARCH Models.

• Student Becar Iuliana
• Supervisor Professor Moisa Altar

2
Table of Contents

• The importance of forecasting exchange rate
  volatility.
• Data description.
• Model estimates and forecasting performances.
• Concluding remarks.

3
Why model and forecast volatility?

• Volatility is one of the most important concepts
  in the whole of finance.
• ARCH models offered new tools for measuring
  risk, and its impact on return.
• Volatility of exchange rates is of importance
  because of the uncertainty it creates for prices
  of exports and imports, for the value of
  international reserves and for open positions in
  foreign currency.

4
Volatility Models.

• ARCH/GARCH models.
• Engle(1982)
• Bollerslev(1986)
• Baillie, Bollerslev and Mikkelsen (1996)
• ARFIMA models.
• Granger (1980)

5
Data description

• Data series nominal daily EUR/ROL exchange rates
• Time length 04011999-11062004
• 1384 nominal percentage returns

6
Descriptive Statistics for the return series.

Statistic t-Test P-Value
Skewness 1.0472 15.922 4.4605e-057
Excess Kurtosis 8.5138 64.769 0.00000
Jarque-Bera 4432.9    
7
Heteroscedasticity
The Daily Return Series
Autocorrelation and Partial autocorrelation of
the Return Series

• The returns are not homoskedastic.
• Low serial dependence in returns.
• The Ljung-Box statistic for 20 lags equals 27.392
  0.125.

8
Autocorrelation and Partial Autocorrelation of
Squared Returns
The Ljung-Box statistic for 20 lags equals
151.010.000
ARCH 1 test 17.955 0.0000 ARCH 2 test
18.847 0.0000
9
Stationarity

• Unit Root Tests for EUR/ROL return series.

ADF Test Statistic -35.60834 1 Critical Value -3.4380
5 Critical Value -2.8641
10 Critical Value -2.5681
MacKinnon critical values for rejection of hypothesis of a unit root. MacKinnon critical values for rejection of hypothesis of a unit root. MacKinnon critical values for rejection of hypothesis of a unit root. MacKinnon critical values for rejection of hypothesis of a unit root.
PP Test Statistic -35.57805 1 Critical Value -3.4380
5 Critical Value -2.8641
10 Critical Value -2.5681
MacKinnon critical values for rejection of hypothesis of a unit root. MacKinnon critical values for rejection of hypothesis of a unit root. MacKinnon critical values for rejection of hypothesis of a unit root. MacKinnon critical values for rejection of hypothesis of a unit root.
10
Model estimates and forecasting performances.

• Methodology.
• Ox Professional 3.30 G_at_RCH4.0
• 4.01.1999-30.12.2002 (1018 observations)
  for model estimation
• 06.01.2003-11.06.2004 (366 observations)
  for out of sample forecast evaluation.
• The Models.
• Two distributions Student, Skewed Student, QMLE.
• The Mean Equations
• 1. A constant mean
• 2. An ARFIMA(1,da,0) mean
• 3. An ARFIMA(0, da,1) mean

11
The variance equations.

• GARCH(1,1) and FIGARCH(1,d,1) without the
  constant term and with a non-trading day dummy
  variable.
• The estimated twelve models.
• Examining the models page 30 to 34 the
  conclusions are
• The estimated coefficients are significantly
  different from zero at the 10 level.
• the ARFIMA coefficient lies between
• which implies stationarity.
• all variance coefficients are positive and

12
In-sample model evaluation. Residual tests. GARCH
models.
Model SBC Skewness EK1 Q Q2 ARCH Nyblom
ARMA (0,0) GARCH(1,1) Skewed-Student 2.210463 0.75224 3.9543 37.5958 0.9019571 30.3204 0.9783154 1.1358 0.3395 1.96933
ARMA (0,0) GARCH(1,1) Student 2.212901 0.74033 3.8319 37.5877 0.9021277 30.3145 0.9783579 1.1238 0.3458 1.58334
ARFIMA (1,d,0) GARCH(1,1) Skewed-Student 2.214579 0.76024 4.1028 36.4188 0.9083405 31.7529 0.9659063 1.2484 0.2843 2.24209
ARFIMA (1,d,0) GARCH(1,1) Student 2.216388 0.73353 3.857 36.0009 0.9165657 31.8411 0.9649974 1.1801 0.3169 1.89543
ARFIMA (0,d,1) GARCH(1,1) Skewed-Student 2.215735 0.75909 4.1153 36.1425 0.9138359 31.3112 0.9701942 1.2084 0.3030 2.2612
ARFIMA (0,d,1) GARCH(1,1) Student 2.217401 0.73390 3.8852 35.8043 0.9202571 31.3087 0.9702172 1.1360 0.3394 1.9047
1 EK-Excess Kurtosis Q-Statistics on
Standardized Residuals with 50 lags
Q-Statistics on Squared Standardized Residuals
50 lags ARCH test with 5 lags P-values in
brackets.
13
In-sample model evaluation. Residual tests.
FIGARCH models.
Model SBC Skewness EK1 Q Q2 ARCH Nyblom
ARMA (0,0) FIGARCH(1,d,1) Skewed-Student 2.222089 0.76305 3.8723 37.4681 0.9046084 28.4572 0.9888560 1.2601 0.2790 1.56799
ARMA (0,0) FIGARCH(1,d,1) Student 2.22472 0.74698 3.7313 37.7303 0.8991133 28.9803 0.9864387 1.3297 0.2491 1.37719
ARFIMA (1,d,0) FIGARCH(1,d,1) Skewed-Student 2.226549 0.757 3.9242 36.3540 0.9096502 29.8994 0.9811947 1.3204 0.2529 2.05757
ARFIMA (1,d,0) FIGARCH(1,d,1) Student 2.228334 0.73378 3.7256 36.1801 0.9131002 30.4315 0.9775013 1.3272 0.2501 1.82764
ARFIMA (0,d,1) FIGARCH(1,d,1) Skewed-Student 2.227516 0.75901 3.96 36.2611 0.9115043 29.2088 0.9852596 1.2729 0.2733 2.0233
ARFIMA (0,d,1) FIGARCH(1,d,1) Student 2.229199 0.73799 3.7813 36.1313 0.9140531 29.5586 0.9832983 1.2630 0.2777 1.79097
1 EK-Excess Kurtosis Q-Statistics on
Standardized Residuals with 50 lags
Q-Statistics on Squared Standardized Residuals
50 lags ARCH test with 5 lags P-values in
brackets.
14
Out-of-sample Forecast Evaluation

• Forecast methodology
• - sample window 1018 observations
• - at each step, the 1 step ahead dynamic
  forecast is stored
• for the conditional variance and the
  conditional mean
• -dynamic forecast is programmed in OxEdit
• G_at_RCH3.0 package
• Benchmark ex-post volatility squared returns.

15
Measuring Forecast Accuracy.

• The Mincer-Zarnowitz regression
• The Mean Absolute Error
• Root Mean Square Error (standard error)
• Theil's inequality coefficient -Theil's U

16
One Step Ahead Forecast Evaluation Measures.
1. The Mincer-Zarnowitz regression
Model alfa beta R2 Model alfa beta R2
ARMA (0,0) GARCH(1,1) Skewed-Student -0.104961 0.0699 0.624769 0.0006 0.0533211 ARMA (0,0) FIGARCH(1,d,1) Skewed-Student -0.038611 0.3070 0.741465 0.0005 0.0822328
ARMA (0,0) GARCH(1,1) Student -0.100843 0.0766 0.617284 0.0007 0.0530545 ARMA (0,0) FIGARCH(1,d,1) Student -0.037921 0.3143 0.725906 0.0005 0.0793558
ARFIMA (1,d,0) GARCH(1,1) Skewed-Student -0.112153 0.0607 0.631864 0.0006 0.0518779 ARFIMA (1,d,0) FIGARCH(1,d,1) Skewed-Student -0.046087 0.2517 0.730264 0.0006 0.0759213
ARFIMA (1,d,0) GARCH(1,1) Student -0.104983 0.0698 0.620363 0.0006 0.0522936 ARFIMA (1,d,0) FIGARCH(1,d,1) Student -0.043940 0.2681 0.707455 0.0006 0.0735089
ARFIMA (0,d,1) GARCH(1,1) Skewed-Student -0.112613 0.0596 0.634110 0.0006 0.052295 ARFIMA (0,d,1) FIGARCH(1,d,1) Skewed-Student -0.045701 0.254 0.731791 0.0006 0.0765561
ARFIMA (0,d,1) GARCH(1,1) Student -0.105667 0.0680 0.623092 0.0006 0.0527494 ARFIMA (0,d,1) FIGARCH(1,d,1) Student -0.043431 0.2715 0.70931 0.0006 0.0742364
17
2. Forecasting the conditional mean. Loss
functions.
Model MAE RMSE TIC Model MAE RMSE TIC
ARMA (0,0) GARCH(1,1) Skewed-Student 0.2601 0.3412 0.7895 ARMA (0,0) FIGARCH(1,d,1) Skewed-Student 0.2606 0.3416 0.7861
ARMA (0,0) GARCH(1,1) Student 0.2576 0.3395 0.812 ARMA (0,0) FIGARCH(1,d,1) Student 0.258 0.3397 0.8086
ARFIMA (1,d,0) GARCH(1,1) Skewed-Student 0.2724 0.3521 0.7527 ARFIMA (1,d,0) FIGARCH(1,d,1) Skewed-Student 0.2726 0.3522 0.7518
ARFIMA (1,d,0) GARCH(1,1) Student 0.2694 0.3493 0.77 ARFIMA (1,d,0) FIGARCH(1,d,1) Student 0.2697 0.3496 0.7684
ARFIMA (0,d,1) GARCH(1,1) Skewed-Student 0.2722 0.352 0.7548 ARFIMA (0,d,1) FIGARCH(1,d,1) Skewed-Student 0.2724 0.3522 0.7536
ARFIMA (0,d,1) GARCH(1,1) Student 0.2691 0.3493 0.7729 ARFIMA (0,d,1) FIGARCH(1,d,1) Student 0.2694 0.3495 0.7711
18
3. Forecasting the conditional variance. Loss
functions.
Model MAE RMSE TIC Model MAE RMSE TIC
ARMA (0,0) GARCH(1,1) Skewed-Student 0.2844 0.3148 0.5253 ARMA (0,0) FIGARCH(1,d,1) Skewed-Student 0.17 0.2234 0.484
ARMA (0,0) GARCH(1,1) Student 0.2824 0.3131 0.5244 ARMA (0,0) FIGARCH(1,d,1) Student 0.1726 0.2253 0.4845
ARFIMA (1,d,0) GARCH(1,1) Skewed-Student 0.2907 0.3204 0.5286 ARFIMA (1,d,0) FIGARCH(1,d,1) Skewed-Student 0.1802 0.2299 0.4856
ARFIMA (1,d,0) GARCH(1,1) Student 0.2866 0.3168 0.5265 ARFIMA (1,d,0) FIGARCH(1,d,1) Student 0.1832 0.2322 0.4861
ARFIMA (0,d,1) GARCH(1,1) Skewed-Student 0.2903 0.32 0.5283 ARFIMA (0,d,1) FIGARCH(1,d,1) Skewed-Student 0.1794 0.2294 0.4854
ARFIMA (0,d,1) GARCH(1,1) Student 0.2862 0.3164 0.5263 ARFIMA (0,d,1) FIGARCH(1,d,1) Student 0.1822 0.2315 0.4859
19
Concluding remarks.

• In-sample analysis
• Residual tests
• -all models may be appropriate.
• -the Student distribution is better
  than the Skewed Student.
• Out-of-sample analysis
• -the FIGARCH models are superior.
• -for the conditional mean the Student
  distribution is
• superior.
• -the two ARFIMA mean equations don't
  provide a better
• forecast of the conditional mean.
  
• - for the conditional variance the
  Skewed Student
• distribution is superior.

20
Concluding remarks.

• Model construction problems
• Further research
• -option prices, which reflect the markets
  expectation
• of volatility over the remaining life
  span of the option.
• -daily realized volatility can be computed
  as the sum of
• squared intraday returns

21
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Bibliography

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23
Appendix 1.
The ARMA (0, 0), GARCH (1, 1) Skewed Student
model. Robust Standard Errors (Sandwich
formula)
Coefficient Std.Error t-value Probability
Constant(Mean) 0.091930 0.021613 4.253 0.0000
dummyFriday (V) 0.048977 0.019781 2.476 0.0134
ARCH(Alpha1) 0.036076 0.011561 3.121 0.0019
GARCH(Beta1) 0.924490 0.018052 51.21 0.0000
Asymmetry 0.145722 0.047250 3.084 0.0021
Tail 9.872213 3.3488 2.948 0.0033
For more details see Appendix 1, page 45.
24
Appendix 2
The ARMA (0, 0), GARCH (1, 1) Student
model. Robust Standard Errors (Sandwich formula)
Coefficient Std.Error t-alue Probability
Constant(Mean) 0.077795 0.021673 3.589 0.0003
dummyFriday (V) 0.049240 0.020163 2.442 0.0148
ARCH(Alpha1) 0.037186 0.011975 3.105 0.0020
GARCH(Beta1) 0.923353 0.018479 49.97 0.0000
Student(DF) 8.921340 2.8119 3.173 0.0016
For more details, see Appendix 2, page 47.
25
Appendix 3
The ARFIMA (1, da, 0),GARCH (1, 1) Skewed
Student model. Robust Standard Errors
(Sandwich formula)
Coefficient Std.Error t-value Probability
Constant(Mean) 0.089939 0.010527 8.544 0.0000
d-Arfima -0.128224 0.045067 -2.845 0.0045
AR(1) 0.123269 0.054553 2.260 0.0241
dummyFriday (V) 0.048860 0.019703 2.480 0.0133
ARCH(Alpha1) 0.033897 0.011677 2.903 0.0038
GARCH(Beta1) 0.926283 0.018096 51.19 0.0000
Asymmetry 0.139771 0.047194 2.962 0.0031
Tail 9.189523 2.9091 3.159 0.0016
For more details, see Appendix 3, page 49.
26
Appendix 4
The ARFIMA (1, da, 0),GARCH (1, 1) Student
model. Robust Standard Errors (Sandwich formula)
Coefficient Std.Error t-value Probabilty
Constant(Mean) 0.082711 0.010237 8.080 0.0000
d-Arfima -0.136317 0.045875 -2.971 0.0030
AR(1) 0.140455 0.055832 2.516 0.0120
dummyFriday (V) 0.049635 0.020117 2.467 0.0138
ARCH(Alpha1) 0.036517 0.012510 2.919 0.0036
GARCH(Beta1) 0.923503 0.018602 49.64 0.0000
Student(DF) 8.436809 2.5257 3.340 0.0009
For more details, see Appendix 4, page 52.
27
Appendix 5
The ARFIMA (0, da,1),GARCH (1, 1) Skewed Student
model. Robust Standard Errors (Sandwich formula)
Coefficient Std.Error t-value Probability
Constant(Mean) 0.090415 0.011041 8.189 0.0000
d-Arfima -0.117757 0.037429 -3.146 0.0017
MA(1) 0.114844 0.046060 2.493 0.0128
dummyFriday (V) 0.048681 0.019787 2.460 0.0140
ARCH(Alpha1) 0.033847 0.011641 2.908 0.0037
GARCH(Beta1) 0.926414 0.018172 50.98 0.0000
Asymmetry 0.138631 0.047049 2.947 0.0033
Tail 9.279306 2.9613 3.134 0.0018
For more details, see Appendix 5, page 54.
28
Appendix 6
The ARFIMA (0, da,1),GARCH (1, 1) Student
model. Robust Standard Errors (Sandwich formula)
Coefficient Std.Error t-value Probability
Constant(Mean) 0.082822 0.010833 7.645 0.0000
d-Arfima -0.122519 0.036843 -3.325 0.0009
MA(1) 0.128311 0.045146 2.842 0.0046
dummyFriday (V) 0.049380 0.020207 2.444 0.0147
ARCH(Alpha1) 0.036344 0.012449 2.919 0.0036
GARCH(Beta1) 0.923788 0.018703 49.39 0.0000
Student(DF) 8.516429 2.5689 3.315 0.0009
For more details, see Appendix 6, page 56.
29
Appendix 7
The ARMA (0, 0), FIGARCH-BBM (1,d,1) Skewed
Student model. Robust Standard Errors (Sandwich
formula)
Coefficient Std.Error t-value Probability
Constant(Mean) 0.094259 0.021931 4.298 0.0000
dummyFriday (V) 0.047278 0.025975 1.820 0.0690
d-Figarch 0.358622 0.098899 3.626 0.0003
ARCH(Alpha1) 0.288896 0.094598 3.054 0.0023
GARCH(Beta1) 0.635309 0.058513 10.86 0.0000
Asymmetry 0.147588 0.046529 3.172 0.0016
Tail 9.545031 3.0964 3.083 0.0021
For more details, see Appendix 7, page 59.
30
Appendix 8
The ARMA (0, 0), FIGARCH-BBM (1,d,1) Student
model. Robust Standard Errors (Sandwich formula)
Coefficient Std.Error t-value Probability
Constant(Mean) 0.079807 0.021915 3.642 0.0003
dummyFriday (V) 0.049310 0.027926 1.766 0.0777
d-Figarch 0.351448 0.10506 3.345 0.0009
ARCH(Alpha1) 0.312018 0.11026 2.830 0.0047
GARCH(Beta1) 0.644842 0.057580 11.20 0.0000
Student(DF) 8.596805 2.6044 3.301 0.0010
For more details, see Appendix 8, page 61.
31
Appendix 9
The ARFIMA (1,da,0), FIGARCH-BBM (1,d,1) Skewed
Student model. Robust Standard Errors (Sandwich
formula)
Coefficient Std.Error t-value Probability
Constant(Mean) 0.090400 0.010719 8.434 0.0000
d-Arfima -0.126724 0.046241 -2.741 0.0062
AR(1) 0.119364 0.054745 2.180 0.0295
dummyFriday (V) 0.052164 0.030787 1.694 0.0905
d-Figarch 0.332074 0.10662 3.115 0.0019
ARCH(Alpha1) 0.339292 0.13642 2.487 0.0130
GARCH(Beta1) 0.649620 0.053779 12.08 0.0000
Asymmetry 0.139501 0.046638 2.991 0.0028
Tail 8.871259 2.6840 3.305 0.0010
For more details, see Appendix 9, page 63.
32
Appendix 10
The ARFIMA (1,da,0), FIGARCH-BBM (1,d,1) Student
model. Robust Standard Errors (Sandwich formula)
Coefficient Std.Error t-value Probability
Constant(Mean) 0.083221 0.010263 8.109 0.0000
d-Arfima -0.136270 0.047181 -2.888 0.0040
AR(1) 0.138494 0.056208 2.464 0.0139
dummyFriday (V) 0.054562 0.034015 1.604 0.1090
d-Figarch 0.328545 0.12291 2.673 0.0076
ARCH(Alpha1) 0.360347 0.17155 2.101 0.0359
GARCH(Beta1) 0.659966 0.057997 11.38 0.0000
Student(DF) 8.093551 2.3226 3.485 0.0005
For more details, see Appendix 10, page 66.
33
Appendix 11
The ARFIMA (0,da,1), FIGARCH-BBM (1,d,1) Skewed
Student model. Robust Standard Errors (Sandwich
formula)
Coefficient Std.Error t-value Probability
Constant(Mean) 0.090938 0.011202 8.118 0.0000
d-Arfima -0.117093 0.039118 -2.993 0.0028
MA(1) 0.112312 0.047184 2.380 0.0175
dummyFriday (V) 0.051724 0.030327 1.706 0.0884
d-Figarch 0.332759 0.10397 3.200 0.0014
ARCH(Alpha1) 0.334340 0.12765 2.619 0.0089
GARCH(Beta1) 0.647135 0.052822 12.25 0.0000
Asymmetry 0.138659 0.046925 2.955 0.0032
Tail 8.973744 2.7438 3.270 0.0011
For more details, see Appendix 11, page 68.
34
Appendix 12
The ARFIMA (0,da,1), FIGARCH-BBM (1,d,1) Student
model. Robust Standard Errors (Sandwich formula)
Coefficient Std.Error t-value Probability
Cst(M) 0.083434 0.010870 7.675 0.0000
d-Arfima -0.122620 0.038276 -3.204 0.0014
MA(1) 0.126887 0.045925 2.763 0.0058
dummyFriday (V) 0.054060 0.033155 1.631 0.1033
d-Figarch 0.329579 0.11765 2.801 0.0052
ARCH(Alpha1) 0.353442 0.15661 2.257 0.0242
GARCH(Beta1) 0.656630 0.055867 11.75 0.0000
Student(DF) 8.182206 2.3695 3.453 0.0006
For more details, see Appendix 12, page 70.
35
Stationarity tests. Appendix 13.
1. Dickey-Fuller Test.
Augmented Dickey-Fuller Test Equation Augmented Dickey-Fuller Test Equation Augmented Dickey-Fuller Test Equation Augmented Dickey-Fuller Test Equation Augmented Dickey-Fuller Test Equation
Dependent Variable D(RETURNS) Dependent Variable D(RETURNS) Dependent Variable D(RETURNS) Dependent Variable D(RETURNS) Dependent Variable D(RETURNS)
Method Least Squares Method Least Squares Method Least Squares Method Least Squares Method Least Squares
Date 06/26/04 Time 0750 Date 06/26/04 Time 0750 Date 06/26/04 Time 0750 Date 06/26/04 Time 0750 Date 06/26/04 Time 0750
Sample(adjusted) 3 1384 Sample(adjusted) 3 1384 Sample(adjusted) 3 1384 Sample(adjusted) 3 1384 Sample(adjusted) 3 1384
Included observations 1382 after adjusting endpoints Included observations 1382 after adjusting endpoints Included observations 1382 after adjusting endpoints Included observations 1382 after adjusting endpoints Included observations 1382 after adjusting endpoints
Variable Coefficient Std. Error t-Statistic Prob.
RETURNS(-1) -0.957262 0.026883 -35.60834 0.0000
C 0.078392 0.018264 4.292148 0.0000
R-squared 0.478843 Mean dependent var Mean dependent var -0.000589
Adjusted R-squared 0.478465 S.D. dependent var S.D. dependent var 0.933223
S.E. of regression 0.673949 Akaike info criterion Akaike info criterion 2.050121
Sum squared resid 626.8057 Schwarz criterion Schwarz criterion 2.057692
Log likelihood -1414.634 F-statistic F-statistic 1267.954
Durbin-Watson stat 1.994863 Prob(F-statistic) Prob(F-statistic) 0.000000
36
ADF Test -17.25675 1 Critical Value -3.4380
5 Critical Value -2.8641
10 Critical Value -2.5681
MacKinnon critical values for rejection of hypothesis of a unit root. MacKinnon critical values for rejection of hypothesis of a unit root. MacKinnon critical values for rejection of hypothesis of a unit root. MacKinnon critical values for rejection of hypothesis of a unit root.
Dependent Variable D(RETURNS) Dependent Variable D(RETURNS) Dependent Variable D(RETURNS) Dependent Variable D(RETURNS) Dependent Variable D(RETURNS)
Method Least Squares Method Least Squares Method Least Squares Method Least Squares Method Least Squares
Sample(adjusted) 7 1384 Sample(adjusted) 7 1384 Sample(adjusted) 7 1384 Sample(adjusted) 7 1384 Sample(adjusted) 7 1384
Included observations 1378 after adjusting endpoints Included observations 1378 after adjusting endpoints Included observations 1378 after adjusting endpoints Included observations 1378 after adjusting endpoints Included observations 1378 after adjusting endpoints
Variable Coefficient Std. Error t-Statistic Prob.
RETURNS(-1) -1.047183 0.060682 -17.25675 0.0000
D(RETURNS(-1)) 0.091319 0.053927 1.693396 0.0906
D(RETURNS(-2)) 0.039379 0.046166 0.852989 0.3938
D(RETURNS(-3)) 0.009635 0.037319 0.258186 0.7963
D(RETURNS(-4)) 0.015333 0.026967 0.568585 0.5697
C 0.086684 0.018835 4.602399 0.0000
R-squared 0.480683 Mean dependent var Mean dependent var 0.000495
Adjusted R-squared 0.478791 S.D. dependent var S.D. dependent var 0.933787
S.E. of regression 0.674146 Akaike info criterion Akaike info criterion 2.053604
Sum squared resid 623.5364 Schwarz criterion Schwarz criterion 2.076369
Log likelihood -1408.933 F-statistic F-statistic 253.9867
Durbin-Watson stat 1.998880 Prob(F-statistic) Prob(F-statistic) 0.000000
Appendix 14. ADF Test.
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Appendix 15.Phillips-Perron Test.
Lag truncation for Bartlett kernel 7 Lag truncation for Bartlett kernel 7 ( Newey-West suggests 7 ) ( Newey-West suggests 7 ) ( Newey-West suggests 7 ) ( Newey-West suggests 7 )
Residual variance with no correction Residual variance with no correction 0.453550 0.453550 0.453550 0.453550
Residual variance with correction Residual variance with correction 0.407637 0.407637 0.407637 0.407637
Phillips-Perron Test Equation Phillips-Perron Test Equation Phillips-Perron Test Equation Phillips-Perron Test Equation Phillips-Perron Test Equation Phillips-Perron Test Equation
Dependent Variable D(RETURNS) Dependent Variable D(RETURNS) Dependent Variable D(RETURNS) Dependent Variable D(RETURNS) Dependent Variable D(RETURNS) Dependent Variable D(RETURNS)
Method Least Squares Method Least Squares Method Least Squares Method Least Squares Method Least Squares Method Least Squares
Sample(adjusted) 3 1384 Sample(adjusted) 3 1384 Sample(adjusted) 3 1384 Sample(adjusted) 3 1384 Sample(adjusted) 3 1384 Sample(adjusted) 3 1384
Included observations 1382 after adjusting endpoints Included observations 1382 after adjusting endpoints Included observations 1382 after adjusting endpoints Included observations 1382 after adjusting endpoints Included observations 1382 after adjusting endpoints Included observations 1382 after adjusting endpoints
Variable Coefficient Coefficient Std. Error t-Statistic Prob.
RETURNS(-1) -0.957262 -0.957262 0.026883 -35.60834 0.0000
C 0.078392 0.078392 0.018264 4.292148 0.0000
R-squared 0.478843 0.478843 Mean dependent var Mean dependent var -0.000589
Adjusted R-squared 0.478465 0.478465 S.D. dependent var S.D. dependent var 0.933223
S.E. of regression 0.673949 0.673949 Akaike info criterion Akaike info criterion 2.050121
Sum squared resid 626.8057 626.8057 Schwarz criterion Schwarz criterion 2.057692
Log likelihood -1414.634 -1414.634 F-statistic F-statistic 1267.954
Durbin-Watson stat 1.994863 1.994863 Prob(F-statistic) Prob(F-statistic) 0.000000